How long is a coastline?

Two people measure the very same wiggly coast — carefully, with rulers. Would they get the same number? You'd think so… right?

1Two things to know first

A coast is wiggly, and a ruler is straight

You need two small ideas. Watch each one happen:

Coasts are wiggly all the way down

Zoom in on a coast and the little bays have tinier bays inside them — bumps on bumps on bumps. The wiggles never run out.

A ruler only does straight

To measure, you lay a straight stick end-to-end and count the steps. But a straight stick has to cut across the wiggles it's too big to fit.

2It depends on your ruler

Same coast — a big ruler and a small ruler

Big ruler 📏

It jumps straight from headland to headland and skips right over the little bays. Fewer steps — a shorter total.

Small ruler 📐

It can dip into each little bay and trace the notches. More steps, more wiggle counted… keep this in mind…

3Walk a ruler around the coast

Pick a ruler and count the steps

Lay coral sticks end-to-end all the way around this island. Slide to pick how long each stick is, and watch the total length on top.

Coast length 0 steps
Ruler length: big
SHORT STICKLONG STICK

A shorter stick fits into more wiggles, so it takes more steps to get around. Notice the total isn't really settling on one number…

4Now predict the shrinking ruler

Keep shrinking the ruler 🔍

Here's the real test. Don't pick one ruler — keep making it smaller and smaller, measuring again each time, hugging tighter into every bay. Where does the total length end up?

Guess before you watch

You measure this coast, then measure again with a much SHORTER ruler, then shorter again. What happens to the total length?